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Mirrors > Home > MPE Home > Th. List > 2sqreuop | Structured version Visualization version GIF version |
Description: There exists a unique decomposition of a prime of the form 4𝑘 + 1 as a sum of squares of two nonnegative integers. Ordered pair variant of 2sqreu 26040. (Contributed by AV, 2-Jul-2023.) |
Ref | Expression |
---|---|
2sqreuop | ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 264 | . . 3 ⊢ ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
2 | 1 | 2sqreu 26040 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (1st ‘𝑝) = (1st ‘〈𝑎, 𝑏〉)) | |
4 | fveq2 6645 | . . . . . 6 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (2nd ‘𝑝) = (2nd ‘〈𝑎, 𝑏〉)) | |
5 | 3, 4 | breq12d 5043 | . . . . 5 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((1st ‘𝑝) ≤ (2nd ‘𝑝) ↔ (1st ‘〈𝑎, 𝑏〉) ≤ (2nd ‘〈𝑎, 𝑏〉))) |
6 | vex 3444 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
7 | vex 3444 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
8 | 6, 7 | op1st 7679 | . . . . . 6 ⊢ (1st ‘〈𝑎, 𝑏〉) = 𝑎 |
9 | 6, 7 | op2nd 7680 | . . . . . 6 ⊢ (2nd ‘〈𝑎, 𝑏〉) = 𝑏 |
10 | 8, 9 | breq12i 5039 | . . . . 5 ⊢ ((1st ‘〈𝑎, 𝑏〉) ≤ (2nd ‘〈𝑎, 𝑏〉) ↔ 𝑎 ≤ 𝑏) |
11 | 5, 10 | syl6bb 290 | . . . 4 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((1st ‘𝑝) ≤ (2nd ‘𝑝) ↔ 𝑎 ≤ 𝑏)) |
12 | 6, 7 | op1std 7681 | . . . . . . 7 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (1st ‘𝑝) = 𝑎) |
13 | 12 | oveq1d 7150 | . . . . . 6 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((1st ‘𝑝)↑2) = (𝑎↑2)) |
14 | 6, 7 | op2ndd 7682 | . . . . . . 7 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (2nd ‘𝑝) = 𝑏) |
15 | 14 | oveq1d 7150 | . . . . . 6 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((2nd ‘𝑝)↑2) = (𝑏↑2)) |
16 | 13, 15 | oveq12d 7153 | . . . . 5 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = ((𝑎↑2) + (𝑏↑2))) |
17 | 16 | eqeq1d 2800 | . . . 4 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → ((((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
18 | 11, 17 | anbi12d 633 | . . 3 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃) ↔ (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
19 | 18 | opreu2reurex 6113 | . 2 ⊢ (∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃) ↔ (∃!𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃!𝑏 ∈ ℕ0 ∃𝑎 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
20 | 2, 19 | sylibr 237 | 1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑝 ∈ (ℕ0 × ℕ0)((1st ‘𝑝) ≤ (2nd ‘𝑝) ∧ (((1st ‘𝑝)↑2) + ((2nd ‘𝑝)↑2)) = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∃!wreu 3108 〈cop 4531 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 1c1 10527 + caddc 10529 ≤ cle 10665 2c2 11680 4c4 11682 ℕ0cn0 11885 mod cmo 13232 ↑cexp 13425 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-phi 16093 df-pc 16164 df-gz 16256 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-imas 16773 df-qus 16774 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-nzr 20024 df-rlreg 20049 df-domn 20050 df-idom 20051 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zn 20200 df-assa 20542 df-asp 20543 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-evls 20745 df-evl 20746 df-psr1 20809 df-vr1 20810 df-ply1 20811 df-coe1 20812 df-evl1 20940 df-mdeg 24656 df-deg1 24657 df-mon1 24731 df-uc1p 24732 df-q1p 24733 df-r1p 24734 df-lgs 25879 |
This theorem is referenced by: (None) |
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